130_notes.dvi

∂L

∂(∂η/∂t)

= μη ̇

−Y

∂^2 η

∂x^2

+μ ̈η+ 0 = 0

η ̈ =

Y

μ

∂^2 η

∂x^2

This is thewave equation for the string. There are easier ways to get to this wave equation, but,
as we move away from simple mechanical systems, a formal way of proceeding will be very helpful.

31.2 Classical Scalar Field in Four Dimensions

Assume we have afield defined everywhere in space and time. For simplicity we will start
with ascalar field(instead of the vector... fields of E&M).

φ(~r,t)

The property that makes this a true scalar field is that it isinvariant under rotations and
Lorentz boosts.
φ(~r,t) =φ′(~r′,t′)

The Euler-Lagrange equation derived from the principle of least action is

∑

k

∂

∂xk

(

∂L

∂(∂φ/∂xk)

)

+

∂

∂t

(

∂L

∂(∂φ/∂t)

)

−

∂L

∂φ

= 0.

Note that since there is only one field, there is only one equation.

Since we are aiming for a description of relativistic quantum mechanics, it will benefit us to write our
equations in acovariant way. I think this also simplifies the equations. We will follow thenotation
of Sakurai. (The convention does not really matter and one should not get hung up on it.) As
usual the Latin indices likei,j,k...will run from 1 to 3 and represent the space coordinates. The
Greek indices likeμ,ν,σ,λ...will run from 1 to 4. Sakurai would give thespacetime coordinate
vectoreither as
(x 1 ,x 2 ,x 3 ,x 4 ) = (x,y,z,ict)

or as
(x 0 ,x 1 ,x 2 ,x 3 ) = (t,x,y,z)

and use the former to do real computations.

We will not use the so called covariant and contravariant indices. Instead we will put anion the
fourth component of a vector which give that component a−sign in a dot product.

xμxμ=x^2 +y^2 +z^2 −c^2 t^2

Note we can have all lower indices. As Sakurai points out, there is noneed for the complication of a
metric tensor to raise and lower indices unless general relativity comes into play and the geometry of
space-time is not flat. We can assume theiin the fourth component is a calculational convenience,
not an indication of the need for complex numbers in our coordinate systems. So while we may
have said “farewell to ict” some time in the past, we will use it here because the notation is less